I used a motor just like that for the scooter I built for my son this summer.
Endless-sphere.com • View topic - Stand up to Mobility scooter conversion it works quite well but I think you will need to run it with a 24v pack it you want it to have enough power to do any good. Also, a controller for it will be 24 volt. Check out the online scooter parts sites. Parts are pretty cheap and since you are in the US shipping shouldn't be too bad. I didn't buy any thing from them because of the high shipping to Canada. If you could find a good deal on 4 more A123 cells, you should be able to stay within your budget. I have the scooter geared really low, but it has lots of torque. I think, even with higher gearing for a bike, it should be fine for a pedal assist if you don't need too much speed. The only problem might be figuring out a way to mount it. I think you will need a second freewheel or something.
Another cheap option might be to pick up an ebike kit off ebay, but I have no idea which sellers are reputable.
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Almost all my driving is done 1-5 miles at a time.
Best short trip: 2.4 l/100 km, 3.9 km